
<template id="subgroups-header-template">
    <p style="width: min(100%, 80em)">All <a href="./help/rf-groupterms/index.html#subgroup">subgroups</a> of
        ${Group.name} are listed in the following table. Also, a link
        which exhibits the embedding for each subgroup is also provided, or in
        some cases a reason why it is not possible right now. Subgroup
        <a href="./help/rf-groupterms/index.html#order-of-a-subgroup">order</a> and
        <a href="./help/rf-groupterms/index.html#index-of-a-subgroup">index</a> are also given.</p>
    <button class="gap-compute" style="grid-column: content / end; margin-left: 1em"
            data-action="ShowGAPCode.setup('getting the list of all subgroups of a group', Group)">Compute this in GAP</button>
    <p style="width: min(100%, 80em)">The column entitled "normal" reports whether the subgroup is
        <a href="./help/rf-groupterms/index.html#normal-subgroup">normal</a>
        in ${Group.name}, and if so, tries to provide a link to
        showing that normality by means of a sheet showing a
        short exact sequence (SES) which exhibits the
        <a href="./help/rf-groupterms/index.html#first-isomorphism-theorem">First Isomorphism Theorem</a>
        applied to the subgroup.</p>
    <button class="gap-compute" style="grid-column: content / end; margin-left: 1em"
            data-action="ShowGAPCode.setup('checking whether a subgroup is normal', Group)">Compute this in GAP</button>
    <p style="width: min(100%, 80em)">The subgroups can also be shown arranged in a
        <a href="./help/rf-groupterms/index.html#lattice-of-subgroups">lattice</a>, each shown as
        highlighted portions of the whole group, connected by the identity (inclusion)
        homomorphism. You may see that lattice by
        <a href="" data-action="SubgroupInfo.showSubgroupLattice('CDElement')">Cayley diagram</a>,
        <a href="" data-action="SubgroupInfo.showSubgroupLattice('CGElement')">cycle graph</a>, or
        <a href="" data-action="SubgroupInfo.showSubgroupLattice('MTElement')">multiplication table</a>, or
        by calculation:</p>
    <button class="gap-compute" style="grid-column: content / end; margin-left: 1em"
            data-action="ShowGAPCode.setup('getting the lattice of subgroups of a group', Group)">Compute this in GAP</button>
    <p id="simple">None of the subgroups on the list below is
        <a href="./help/rf-groupterms/index.html#normal-subgroup">normal</a>.
        For this reason, ${Group.name} is a
        <a href="./help/rf-groupterms/index.html#simple-group">simple</a> group.</p>
    <p id="not-simple">At least one of the subgroups on the list below is
        <a href="./help/rf-groupterms/index.html#normal-subgroup">normal</a>.
        For this reason, ${Group.name} is not a
        <a href="./help/rf-groupterms/index.html#simple-group">simple</a> group.</p>
    <button class="gap-compute" style="grid-column: content / end; margin-left: 1em"
            data-action="ShowGAPCode.setup('checking if a group is simple', Group)">Compute this in GAP</button>
    <p></p>
    <div style="overflow: hidden; max-width: 80em; width: fit-content; width: -moz-fit-content; touch-action: none; border: 3px solid silver;">
        <table style="border: 0; border-spacing: 0; border-collapse: collapse; padding: 0;">
            <thead style="border-bottom: 3px solid silver; display: table; overflow-y: auto; width: 100%;">
                <tr>
                    <th style="border: 0; padding: 4px 0; text-align: center; width: 20%">Image</th>
                    <th style="border: 0; padding: 4px 0; text-align: center; width: 20%">Name</th>
                    <th style="border: 0; padding: 4px 0; text-align: center; width: 20%">Order</th>
                    <th style="border: 0; padding: 4px 0; text-align: center; width: 20%">Index</th>
                    <th style="border: 0; padding: 4px 0; text-align: center; width: 20%">Normal</th>
                </tr>
            </thead>
            <tbody style="display: block; max-height: 60em; overflow-y: auto; width: 100%;">
            </tbody>
        </table>
    </div>
    <p id="subgroups-no-isomorphism-reason">* No group
        <a href="./help/rf-groupterms/index.html#isomorphism-isomorphic">isomorphic</a>
        to this subgroup appears in the group library. This makes it
        impossible to draw embedding diagrams, or examples of the
        <a href="./help/rf-groupterms/index.html#first-isomorphism-theorem')">First Isomorphism Theorem</a>.
        You can choose what groups to include in your group library in the
        <a href="">options window</a>.</p>
    <p id="subgroups-no-quotient-group-reason">** No group
        <a href="./help/rf-groupterms/index.html#isomorphism-isomorphic">isomorphic</a>
        to the quotient of ${Group.name} by the normal subgroup appears
        in the group library. This makes it impossible to illustrate the quotient
        group. You can choose what groups to include in your group library in the
        <a href="">options window</a>.</p>
</template>

<template id="subgroups-data-row-template">
    <tr style="border-top: 2px solid silver;">
        <td class="image" style="border: 0; border-right: 1px solid silver; padding: 2px 3px 3px 4px; width: 20%;">image</td>
        <td style="border: 0; border-right: 1px solid silver; padding: 2px 3px 3px 4px; width: 20%;"><i>H</i><sub>${index}</sub></td>
        <td style="border: 0; border-right: 1px solid silver; padding: 2px 3px 3px 4px; width: 20%;">${subgroup.order}</td>
        <td style="border: 0; border-right: 1px solid silver; padding: 2px 3px 3px 4px; width: 20%;">${subgroup.index}</td>
        <td style="border: 0; padding: 1px 3px 3px 4px; width: 20%;">${Group.isNormal(subgroup) ? 'yes' : 'no'}</td>
    </tr>
    <tr style="border-top: 1px solid silver">
        <td colspan="5" style="border: 0; padding: 2px 3px 3px 4px">
            <p class="intro"><i>H</i><sub>${index}</sub>${optionalDescription} is
                { ${element_representations.join(', ')} }.</p>
            <ul class="points">
            </ul>
        </td>
    </tr>
</template>

<template id="subgroups-isomorphism-template">
    <li>It is
        <a href="./help/rf-groupterms/index.html#isomorphism-isomorphic">isomorphic</a> to
        ${isomorphicGroup.name}, and you can see the embedding by
        <a href="" data-action="SubgroupInfo.showEmbeddingSheet(${index}, 'CDElement')">Cayley diagram</a>,
        <a href="" data-action="SubgroupInfo.showEmbeddingSheet(${index}, 'CGElement')">cycle graph</a>,
        <a href="" data-action="SubgroupInfo.showEmbeddingSheet(${index}, 'MTElement')">multiplication table</a>,
    </li>
</template>

<template id="subgroups-no-isomorphism-template">
    <li class="subgroups-no-isomorphism">I cannot show you the embedding
        right now (see <a href="#subgroups-no-isomorphism-reason">below*</a>).
    </li>
</template>

<template id="subgroups-quotient-group-template">
    <li >See the <a href="./help/rf-groupterms/index.html#short-exact-sequence">short exact sequence</a>
        exhibiting the
        <a href="./help/rf-groupterms/index.html#quotient-group">quotient group</a>,
        isomorphic to ${isomorphicGroup.name}, by
        <a href="" data-action="SubgroupInfo.showQuotientSheet(${index}, 'CDElement')">Cayley diagram</a>,
        <a href="" data-action="SubgroupInfo.showQuotientSheet(${index}, 'CGElement')">cycle graph</a>,
        <a href="" data-action="SubgroupInfo.showQuotientSheet(${index}, 'MTElement')">multiplication table</a>,
    </li>
</template>

<template id="subgroups-no-quotient-group-template">
    <li class="subgroups-no-quotient-group">I cannot show you the quotient group
        right now (see <a href="#subgroups-no-quotient-group-reason">below**</a>).
    </li>
</template>
